Of course if in-doubt be sure to call a professional HVAC technician.
A new motor and capacitor will need to be bought to replace the old unit.
Most hardware store will not carry a furnace motor but a new unit can be bought easily online on Amazon or eBay.
Example Furnace Blower Motor on Amazon
1/3 hp 1075 RPM 3-Speed 48 Frame 115V Direct Drive Furnace Motor EM3585
The first step is to locate the motor which is the only large motor in the furnace.
Also look for a Start or RUN Capacitor since this will also have to be changed (if the furnace has one). A bad or failing capacitor can damage a new motor so it must be changed any time a new motor is installed.
Some furnaces don’t have a RUN or Start capacitor which can vary between units.
Be sure to turn off the power to the furnace or unplug it before any work is done to a motor or any components.
Also before anything is removed or any work done take pictures and write down all the wiring and connections.
Take good pictures or video to refer back to later if needed. If something is wired wrong the motor may burn up or at the very least not work. Having pictures to look back at can be invaluable when reconnecting everything.
Once you have a good pictures of the wiring, the wires can be unplugged and removed.
The motor will be inside of a sheet metal case making it necessary to remove the entire assembly. Screws holding it in are usually located in the front with a notch in the back it can slide out of.
Once the entire assembly is taken out the bolts holding it in place need to be removed. One bolt will hold onto the shaft to turn the fan. To remove it loosen it up so that the shaft can pull out from the fan blades. It is a good idea to oil up the end of the shaft so it pulls out easier as some can be a pain. If there is any rust or build up on the shaft use sandpaper to remove it then oil it. Otherwise pulling a motor from the fan blades can be very hard sometimes.
Next the bolts holding the back of the blower motor should be removed. And the motor pulled from the fan blades. (On a side note it is a good idea to clean the fan blades when the assembly is out. Debris build up on fan blades make them work less efficiently cleaning them is always a good idea. If the fan blades are really dirty take them to a car wash and after spraying them with degreaser power wash them.)
The motor will have a ring that goes around it and bolts to the housing, this should be taken off the old motor and put one the new one. In some cases is can be damaged if it has been a new one can be bought.
Example Replacement Furnace Motor Bracket On Amazon.
Dayton Motor Mtg Bracket, 3-Ring, Anodized Steel – 2MEV2
Once the ring has been place on the new motor the shaft can be slide back on to the fan blades, and bolted into place.
When bolting the fan blades in they will need to be centered so the blades spin freely in the housing.
Before tightening the bolt on the shaft be sure it is centered correctly since there is a flat spot on the shaft that is where the bolt screws into holding it in place.
Before installing the assembly back into the furnace be sure the fan blades spin without hitting the housing.
Connect the ground wire.
Install the new capacitor and connect the correct wires from the motor.
Double check everything is connected and the fan blades are spinning correctly and it will be time to reinstall the assembly back into the furnace.
Installing will simply be the reverse of taking it out. The hardest part is often getting it to slide in correctly into the metal tabs.
After getting it back into place tighten it in with the holding screws.
After getting the assembly back in double check the fan blades again for proper spin. You don’t want to damage anything when power is applied and they spin fast.
After the assembly is back into place the wiring can be reconnected. This is where the picture taken earlier come into play.
Here is what the colors are for in most cases. Some units may be different.
White wire is for the blower
Black wire is for wire cooling
Blue wire is for wire heating
Red wire is connects to spare
If there is a problem look at the schematic for the furnace.
Turn the power back on. It is always a good idea to double check everything before turning the power back on.
Once the power is back on and the thermostat set the furnace should kick on and blower motor begin to spin.